Normal not implies normal-extensible automorphism-invariant in finite

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., normal-extensible automorphism-invariant subgroup)
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Statement

Statement with symbols

It is possible to have a finite group G, a normal subgroup N of G, and a normal-extensible automorphism \sigma of G such that \sigma(N) \ne N.

Related facts

Weaker facts

Applications

Facts used

  1. Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
  2. Automorphism group of direct power of simple non-abelian group equals wreath product of automorphism group and symmetric group

Proof

Example of the dihedral group

Further information: dihedral group:D8, subgroup structure of dihedral group:D8

Let G be the dihedral group of order eight. Then, every automorphism of G fixes every element of the center of G, and also, the inner automorphism group of G is maximal in the automorphism group of G. Thus, by fact (1), every automorphism of G is normal-extensible.

However, there is an automorphism of G that interchanges the two normal Klein four-subgroups. Thus, these two normal subgroups are not invariant under this automorphism, and hence, we have an automorphism of G that is normal-extensible but not normal.

Equivalently, the Klein four-subgroups are examples of normal subgroups that are not normal-extensible automorphism-invariant.

Example involving a simple complete group

Let S be a simple complete group. In other words, S is a centerless simple group such that every automorphism of S is inner. Let G = S \times S. By fact (2), the automorphism group of G is the wreath product of S with the symmetric group of degree two, which has G, the inner automorphism group, as a subgroup of index two. Moreover, G is centerless. Thus, by fact (1), we get that every automorphism of G is normal-extensible.

However, the coordinate exchange automorphism of G, that interchanges the two copies of S, is not a normal automorphism because it interchanges these two normal subgroups. Thus, we have an example of a normal-extensible automorphism that is not normal.

Equivalently, either of the direct factors is an example of a normal subgroup that is not normal-extensible automorphism-invariant.